Art and design are both very similar but they still have their differences. The major difference between art and design is that art is more about self expression and having a deeper meaning behind the art work. Design can be similar since there are graphic designers who put a deeper meaning behind their art work. However majority of the time, design usually has a purpose and that purpose is to persuade and push the consumer to do or buy something like advertising or branding.

Art Example:  https://www.flickr.com/photos/gerardolino/5777209930/

Design Example: http://digitalmash.com/#flavors.me